Wavelets: Principles and Applications in Signal Processing. Wavelets: Principles and Applications in Signal Processing

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Miles Harmon
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1 Wavelets: Principles and Applications in Signal Processing Instructor: Dr. Hossain Ahmadi Noubari Office: Room 305 ECE building Phone: ? Prerequisite: An undergraduate course on signals and systems and signal processing An outline of a graduate course on wavelets: Wavelets: Principles and Applications in Signal Processing A Note on Wavelets: Wavelets are powerful signal processing tools that have found applications in a broad spectrum of scientific and applied engineering problems. Some of the current engineering applications include image processing, communication, data storage and compression as well as information extraction for pattern recognition and diagnostics. Concepts and theories of wavelets provide a unified framework for a number of technologies developed independently for various signal processing applications including filter banks, multi-resolution and subspace analysis. Topical Outline I. Preliminaries on Signal Processing: Motivations, Concepts and Transforms, 1. Signal processing, preliminaries on Fourier transform, illustrative Examples From Fourier transform to windowed STFT, Gabor and wavelet transform, time and frequency analysis, phase plane representations, Why wavelets: Localization by wavelets, decorrelation by wavelets, transient nonstationary data analysis, time-frequency localization, signal separation Wavelets in signal processing, background information Wavelets as basis function, atoms for signal decomposition, analysis and function approximation, wavelet as optimal basis functions Characterization of wavelets: Finite duration, oscillatory behavior conditions for signal reconstruction, admissibility condition Illustrative examples of several wavelets: Daubechies, Symlet, Coiflet, Biorthogonal, Haar Morlet wavelet 2. Main stages in signal processing: Signal decomposition, analysis stage Signal processing stage for information extraction, for communication, coding and storage, Signal reconstruction and synthesis stage 1

2 3. Function spaces and transform analysis, an overview 4. Inner product spaces, subspace analysis, Riezs bases Orthogonal, biorthogonal and redundant transform Scaling and translation in wavelet analysis for basis generation, mother Wavelet Continuous time and discrete-time wavelets transform(dwt) Nested subspace analysis in DWT, dyadic transform Scaling and wavelet functions Refinement(dilation, interpolation) equation in standard DWT 5. Wavelet transform of L^2 functions Scaling and wavelet transform, scaling and wavelet coefficients Physical interpretation of wavelet transform and coefficients 6. Illustrative examples of wavelet transform of selected signals 7. Use of Matlab Wavemenu toolbox, other software packages and downloads II- Mathematics of Wavelets and Wavelet Analysis, Basic Concepts 1. Wavelet transform, definition, Morlet Grossman wavelets, continuous and discrete-time wavelet transform 2. Wavelets as optimal basis function in signal decomposition and function approximation, wavelets as unconditional bases( Donoho) 3. Theorems Linearity of wavelet transform Similarity theorem Shift theorem, shift variant and invariant wavelet transforms Differentiation theorem Convolution theorem 4. Redundancy in transform, Redundancy in CWT, frames, definition and derivations, Sylvester inequality, order of redundancy and frames, compact frame 5. Nyquist sampling and theoretical dimension of a function, scaling coefficients and sample values III- Subsampling and Upsampling, Polyphase Representation and Signal Analysis Decimation in wavelet transform, convolution followed by subsampling Subsampling and upsampling in time domain Nobel identity theorems, Nobel Identity 1 and 2 Down-sampling and upsampling in z-domain and in frequency domain Polyphase representation of a signal, odd and even samples Polyphase analysis in decimation by 2 and M, polyphase for interpolation 2

3 Polyphase representation in z-domain Matrix representation of polyphase analysis Standard implementation, polyphase implementation VI- Filter Banks in Wavelet Analysis 1. Why filter banks, Filter bank implementation of two and M band wavelet signal decomposition and reconstruction 2. Analysis and synthesis banks 2 band and M band filter bank 3. Perfect reconstruction and alias elimination in filter banks, Conditions for signal reconstruction and alias elimination Condition for realizability of signal reconstruction 4. Performance analysis of filter banks, analysis and synthesis banks Distortion, aliasing, phase and amplitude distortion 5. Matrix representation in time and z domain, polyphase analysis Condition for distortion elimination and reconstruction Linear phase filter bank construction 6. Design alternatives for construction of low/high pass filters 7. Orthogonal FIR and Biorthogonal IIR filter banks 8. Illustrative Examples. Orthogonal. Biorthogonal, FIR filters and wavelets IV- Multiresolution Analysis, Mallat Algorithm 1. From refinement equation to multiresolution signal analysis, FIR filters for multiresolution wavelet analysis 2. Refinement equation in DWT and construction of scaling and wavelet functions from low/high-pass filters 3. Iterative algorithm for construction of scaling and wavelet functions in time and frequency domain, 4. Multiresolution analysis and subspace interpretation and completeness 5. Mallat algorithm for wavelet construction and software for wavelet design 6. Illustrative examples V- Properties of Wavelets and Wavelet Analysis, and Edge Effects 1. Regularity, linear phase, moment cancellation, wavelet order, frequency selectivity, design of maximally flat Daubechies filters 2. Finite length signals, edge effect in wavelet analysis, periodic wavelets, periodic wavelet analysis, alternative schemes for removal of edge effects VI- Wavelet packets 1. Wavelet packets and a narrow frequency band analysis, 2. Wavelet packet binary tree and signal decomposition 3

4 3. Wavelet packet dictionary, redundancy in wavelet packet analysis 4. Best basis and best tree, alternative cost functions: Entropy-based cost function for compression Informative wavelets Local discriminatory basis selection Others( log energy, coefficient threshold, L p norm concentration. VII- Wavelet Dictionaries, Algorithms for Wavelet Selection 1. Best basis selection 2. Matching Pursuit 3. Best Pursuit 4. Projection pursuit VIII- Denoising, Wavelet Shrinkage 1. Wavelets as decorrelators, regularity and signal smoothness, 2. Additive noise model, white noise model and universal thresholding using high frequency details coefficients 3. Nonwhite noise, level dependent noise model and universal thresholding 4. Signal and noise level estimation, minimax criterion and threshold estimation rule 5. Alternative thresholding schemes( soft and hard) and selection rules 6. Hidden Markov noise models in wavelet analysis IX- Data Compression by Wavelets ( Introduction only) 1. Basic idea: Signal transformation into domains with sparse representation, entropy criteria for wavelet selection 2. Brief reference to alternative methods of data compression, lossless and lossy compression ( DCT, PCA-KLT, JPEG) 3. Integer to integer transform, lossless Compression, lifting scheme X- Redundant Wavelet Transform, Stationary Wavelet Transform (SWT) XI- Applications ( Illustrative application examples will be given, some will be assigned as student projects). Areas for possible applications are as follows: Wavelets in biomedical signal analysis for feature extraction (EEG, ECG, radiographic X-rays, MRI images), Wavelets in medical applications for image enhancement, contrast amplification, edge detection, Wavelets in noise reduction applications, Wavelets in system identification and control, Wavelets in industrial machine applications and diagnosis(me students), Wavelets in watermarking Wavelets and data fusion, 4

5 Wavelets for image and data compression ( JPEG 2000). Wavelets in seismic signal analysis and mining (for CE students) 5

Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)

Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let) Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2π-periodic square-integrable function is generated by a superposition